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Explicit Noether Lefschetz for arbitrary threefolds

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27 Pages
English

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Explicit Noether-Lefschetz for arbitrary threefolds ANGELO FELICE LOPEZ1 and CATRIONA MACLEAN2 Abstract We study the Noether-Lefschetz locus of a very ample line bundle L on an arbitrary smooth threefold Y . Building on results of Green, Voisin and Otwinowska, we give explicit bounds, depending only on the Castelnuovo- Mumford regularity properties of L, on the codimension of the components of the Noether-Lefschetz locus of |L| . 1 Introduction. It is well-known in algebraic geometry that the geometry of a given variety is influenced by the geometry of its subvarieties. It less common, but not unusual, that a given ambient variety forces to some extent the geometry of its subvarieties. A particularly nice case of the latter is given by line bundles, whose properties do very much influence the geometry. If Y is a smooth variety and i : X ?? Y is a smooth divisor, there is then a natural restriction map i? : Pic(Y ) ? Pic(X) given by pull-back of line bundles. Now suppose that X is very ample. By the Lefschetz theorem i? is injective if dimY ≥ 3. On the other hand, it was already known to the Italian school (Severi [18], Gherardelli [6]), that i? is surjective when dimY ≥ 4. Simple examples show that in the case where dimY = 3 we cannot hope for surjectivity unless a stronger restriction is considered.

  • p2-bundle then

  • ?y ?

  • ample line

  • geometria delle

  • codim ?l

  • noether-lefschetz locus

  • line bundles


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Explicit
1
Noether-Lefschetz
for arbitrary threefolds
ANGELO FELICE LOPEZ1and CATRIONA MACLEAN2
Abstract WestudytheNoether-LefschetzlocusofaveryamplelinebundleLon an arbitrary smooth threefoldY on results of Green, Voisin and. Building Otwinowska, we give explicit bounds, depending only on the Castelnuovo-Mumford regularity properties ofL, on the codimension of the components oftheNoether-Lefschetzlocusof|L|.
Introduction.
Itiswell-knowninalgebraicgeometrythatthegeometryofagivenvarietyis influenced by the geometry of its subvarieties. It less common, but not unusual, that a given ambient variety forces to some extent the geometry of its subvarieties. A particularly nice case of the latter is given by line bundles, whose properties do very much influence the geometry. IfYis a smooth variety andi:X ,Yis a smooth divisor, there is then a natural restriction map i: Pic(Y)Pic(X) givenbypull-backoflinebundles. Now suppose thatX Byis very ample. the Lefschetz theoremiis injective if dimY3 the other hand, it was already known to the Italian school (Severi. On [18], Gherardelli [6]), thatiis surjective whendimY4. Simple examples show that in the case wheredimY= 3we cannot hope for surjectivity unless a stronger restriction is considered. For the caseY=P3classical problem, first posed by Noether and, this is also a solved in the case of genericXby Lefschetz who showed that
Theorem(Noether-Lefschetz)ForXa generic surface of degreed4inP3 we havePic(X)=Z. Here and below by generic we mean outside a countable union of proper sub-varieties. 1the MIUR national project “Geometria delle varieta` alge-Research partially supported by briche” COFIN 2002-2004 and by the INdAM project “Geometria birazionale delle variet `a alge-briche”. 2Research partially supported by ???.
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Suppose now that a smooth threefoldYand a line bundleLonYneW.eariveg willsaythataNoether-Lefschetztheoremholdsforthepair(Y L), if i: Pic(Y)Pic(X)
is a surjection for a generic smooth surfaceXYsuch thatOY(X) =L. The following result of Moishezon ([14], see also the argument given in Voisin [21,Thm.15.33])establishestheexactconditionsunderwhichaNoether-Lefschetz theorem holds for(Y L).
Theorem (Moishezon)If(Y L)are such thatLis very ample and h0,ve2(XC)6= 0 for a generic smoothXsuch thatOY(X) =LoehtmerenaN,threL-eohtehztfecs holds for the pair(Y L). Here,h0,ev2denotes the evanescent(20)-cohomology ofX below for a pre-: see cise definition.
More precisely, we denote byU(L)the open subset ofPH0(L)parameteriz-ing smooth surfaces in the same equivalence class asL. We further denote by NL(L)(oNehehteeL-rhcsftfocosutelzL) the subspace parameterizing surfaces Xbunelithwiedppuiqefkcamorbypull-bproducedhcranetodnelwsihY. The above theorem then admits the following alternative formulation.
Theorem (Moishezon)If(Y L)are such thatLis very ample and h0,ve2(XC)6= 0 for a generic smoothXsuch thatOY(X) =LsolucNehtneht,tzhescef-Lerthoe NL(L)is a countable union of proper algebraic subvarieties ofU(L).
These proper subvarieties will henceforth be referred to ascomponents of the Noether-Lefschetzlocus.
ANoether-Lefschetztheoremforapair(Y L)essentially says that for a generic surfaceXsuch thatOY(X) =L, the set of line bundles onXis well-understood andassimpleaspossible.Anaturalfollow-upquestionis:howrarearesurfaces with badly behaved Picard groups? Or alternatively: how large can the compo-nentsoftheNoether-LefschetzlocusbeincomparisonwithU(L) leads us? This
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